The ratio of ages of the father and mother was `11 : 10` when their son was bom. The ratio of ages of the father and mother will be `19 : 18` when the son will be twice his present age. What is the ratio of present ages of father and mother ?

Options

A. `15 : 14`
B. `14 : 13`
C. `16 : 15 `
D. `17 : 16`

Answer

To solve the problem step by step, we will denote the present ages of the father, mother, and son as F, M, and C respectively.

### Step 1: Establish the initial ratio of ages when the son was born
When the son was born, the ratio of the father's age to the mother's age was given as 11:10. This means:
\[ \frac{F - C}{M - C} = \frac{11}{10} \]
Where \(C\) is the age of the son at the time of birth (which is 0 years). Therefore, we can express this as:
\[ 10(F - C) = 11(M - C) \]
This simplifies to:
\[ 10F - 10C = 11M - 11C \]
Rearranging gives us:
\[ 10F - 11M = -C \]
(Equation 1)

### Step 2: Establish the ratio of ages when the son is twice his present age
We are told that when the son is twice his present age, the ratio of the father's age to the mother's age will be 19:18. This means:
\[ \frac{F + C}{M + C} = \frac{19}{18} \]
This can be expressed as:
\[ 18(F + C) = 19(M + C) \]
Simplifying this gives:
\[ 18F + 18C = 19M + 19C \]
Rearranging gives us:
\[ 18F - 19M = C \]
(Equation 2)

### Step 3: Solve the equations
Now we have two equations:
1. \(10F - 11M = -C\) (Equation 1)
2. \(18F - 19M = C\) (Equation 2)

We can add these two equations:
\[ 10F - 11M + 18F - 19M = -C + C \]
This simplifies to:
\[ 28F - 30M = 0 \]
Rearranging gives us:
\[ 28F = 30M \]
Dividing both sides by 2 gives:
\[ 14F = 15M \]
Thus, the ratio of the present ages of the father and mother is:
\[ \frac{F}{M} = \frac{15}{14} \]

### Step 4: Conclusion
The ratio of the present ages of the father and mother is \(15:14\).